

respectively, where \(\left<\cdot ,\cdot \right>\)  denotes the standard inner (dot) product on \(\mathbb {R}^{4}\) .
To approximate the results of an undiscounted game, we use a discounted game with a small discounting rate, \(\lambda =1-10^{-3}\) . In this way, we ensure that the limiting payoffs always exist, even if the players adopt strategies that allow for multiple absorbing states, such as tit-for-tat [1]}.
